Well-posedness of the mean field forest fire age evolution equation

Abstract

We prove the well-posedness of a differential equation that describes the evolution of the large-system limit of the empirical age measure in the mean field forest fire model of Ráth and Tóth (arXiv:0808.2116). This forest fire model is a random graph process on $n$ vertices, whose dynamics combine the Erdős-Rényi dynamics with a Poisson rain of lightning strikes. All edges in any connected component are deleted as soon as any of its vertices is struck by lightning. Each vertex has an age, which increases at rate $1$ but is reset to $0$ each time it burns. We consider the asymptotic lightning regime in which the model displays self-organized criticality. Crane, Ráth and Yeo (arXiv:1811.07981) take the initial state to be an inhomogeneous random graph whose edge probabilities depend on the ages of the vertices. They show that as $n \to \infty$ the empirical age distribution converges as a process to the solution of a deterministic autonomous differential equation. It is a nonlinear age-dependent population dynamics model whose age-specific mortality modulus involves the leading eigenfunction of the branching operator of an associated multitype branching process. The differential equation displays self-organized criticality in the sense that the leading eigenvalue of the branching operator is held at $1$ without this being imposed as a boundary condition.